TY - CHAP
T1 - Degree
AU - Brezis, Haïm
AU - Mironescu, Petru
N1 - Funding Information:
ACKNOWLEDGEMENT This study was supported by the Netherlands Foundation for Chemical Research (SON) with financial aid form from the Netherlands Organization for the Advancement of Pure Research (ZWO).
Publisher Copyright:
© 2021, Springer Science+Business Media, LLC, part of Springer Nature.
PY - 2021
Y1 - 2021
N2 - We revisit the notion of topological degree deg f (aka index or winding number) for maps f: S1→ S1. This is a classical concept when f is continuous: deg f counts “how many times f(S1) covers S1, taking into account algebraic multiplicity.” One can still give a robust definition for deg f when f belongs merely to VMO(S1;S1), and thus, by the Sobolev embeddings, for maps in the critical spaces W1/p,p(S1; S1), with 1 < p< ∞. We establish some basic properties of this degree.
AB - We revisit the notion of topological degree deg f (aka index or winding number) for maps f: S1→ S1. This is a classical concept when f is continuous: deg f counts “how many times f(S1) covers S1, taking into account algebraic multiplicity.” One can still give a robust definition for deg f when f belongs merely to VMO(S1;S1), and thus, by the Sobolev embeddings, for maps in the critical spaces W1/p,p(S1; S1), with 1 < p< ∞. We establish some basic properties of this degree.
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U2 - 10.1007/978-1-0716-1512-6_12
DO - 10.1007/978-1-0716-1512-6_12
M3 - Chapter
AN - SCOPUS:85122460510
T3 - Progress in Nonlinear Differential Equations and Their Application
SP - 339
EP - 380
BT - Progress in Nonlinear Differential Equations and Their Application
PB - Birkhauser
ER -